3.556 \(\int \frac{x^{-1+\frac{n}{4}}}{a+b x^n+c x^{2 n}} \, dx\)

Optimal. Leaf size=353 \[ \frac{2\ 2^{3/4} c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{n \sqrt{b^2-4 a c} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{2\ 2^{3/4} c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{n \sqrt{b^2-4 a c} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}} \]

[Out]

(2*2^(3/4)*c^(3/4)*ArcTan[(2^(1/4)*c^(1/4)*x^(n/4))/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(Sqrt[b^2 - 4*a*c]*(-b -
Sqrt[b^2 - 4*a*c])^(3/4)*n) - (2*2^(3/4)*c^(3/4)*ArcTan[(2^(1/4)*c^(1/4)*x^(n/4))/(-b + Sqrt[b^2 - 4*a*c])^(1/
4)])/(Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*n) + (2*2^(3/4)*c^(3/4)*ArcTanh[(2^(1/4)*c^(1/4)*x^(n/4
))/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*n) - (2*2^(3/4)*c^(3/4)*
ArcTanh[(2^(1/4)*c^(1/4)*x^(n/4))/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])
^(3/4)*n)

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Rubi [A]  time = 0.62696, antiderivative size = 353, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {1381, 1347, 212, 208, 205} \[ \frac{2\ 2^{3/4} c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{n \sqrt{b^2-4 a c} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{2\ 2^{3/4} c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}+\frac{2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{n \sqrt{b^2-4 a c} \left (-\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{n \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + n/4)/(a + b*x^n + c*x^(2*n)),x]

[Out]

(2*2^(3/4)*c^(3/4)*ArcTan[(2^(1/4)*c^(1/4)*x^(n/4))/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(Sqrt[b^2 - 4*a*c]*(-b -
Sqrt[b^2 - 4*a*c])^(3/4)*n) - (2*2^(3/4)*c^(3/4)*ArcTan[(2^(1/4)*c^(1/4)*x^(n/4))/(-b + Sqrt[b^2 - 4*a*c])^(1/
4)])/(Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*n) + (2*2^(3/4)*c^(3/4)*ArcTanh[(2^(1/4)*c^(1/4)*x^(n/4
))/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(Sqrt[b^2 - 4*a*c]*(-b - Sqrt[b^2 - 4*a*c])^(3/4)*n) - (2*2^(3/4)*c^(3/4)*
ArcTanh[(2^(1/4)*c^(1/4)*x^(n/4))/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])
^(3/4)*n)

Rule 1381

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/(m + 1), Subst[Int[(a +
b*x^Simplify[n/(m + 1)] + c*x^Simplify[(2*n)/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, c, m, n, p}, x
] && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IntegerQ[Simplify[n/(m + 1)]] &&  !IntegerQ[n]

Rule 1347

Int[((a_) + (b_.)*(x_)^(n_) + (c_.)*(x_)^(n2_))^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[c/q, In
t[1/(b/2 - q/2 + c*x^n), x], x] - Dist[c/q, Int[1/(b/2 + q/2 + c*x^n), x], x]] /; FreeQ[{a, b, c}, x] && EqQ[n
2, 2*n] && NeQ[b^2 - 4*a*c, 0]

Rule 212

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-(a/b), 2]], s = Denominator[Rt[-(a/b), 2]
]}, Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&
!GtQ[a/b, 0]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^{-1+\frac{n}{4}}}{a+b x^n+c x^{2 n}} \, dx &=\frac{4 \operatorname{Subst}\left (\int \frac{1}{a+b x^4+c x^8} \, dx,x,x^{n/4}\right )}{n}\\ &=\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx,x,x^{n/4}\right )}{\sqrt{b^2-4 a c} n}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^4} \, dx,x,x^{n/4}\right )}{\sqrt{b^2-4 a c} n}\\ &=\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-b-\sqrt{b^2-4 a c}}-\sqrt{2} \sqrt{c} x^2} \, dx,x,x^{n/4}\right )}{\sqrt{b^2-4 a c} \sqrt{-b-\sqrt{b^2-4 a c}} n}+\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-b-\sqrt{b^2-4 a c}}+\sqrt{2} \sqrt{c} x^2} \, dx,x,x^{n/4}\right )}{\sqrt{b^2-4 a c} \sqrt{-b-\sqrt{b^2-4 a c}} n}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-b+\sqrt{b^2-4 a c}}-\sqrt{2} \sqrt{c} x^2} \, dx,x,x^{n/4}\right )}{\sqrt{b^2-4 a c} \sqrt{-b+\sqrt{b^2-4 a c}} n}-\frac{(4 c) \operatorname{Subst}\left (\int \frac{1}{\sqrt{-b+\sqrt{b^2-4 a c}}+\sqrt{2} \sqrt{c} x^2} \, dx,x,x^{n/4}\right )}{\sqrt{b^2-4 a c} \sqrt{-b+\sqrt{b^2-4 a c}} n}\\ &=\frac{2\ 2^{3/4} c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4} n}-\frac{2\ 2^{3/4} c^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \left (-b+\sqrt{b^2-4 a c}\right )^{3/4} n}+\frac{2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \left (-b-\sqrt{b^2-4 a c}\right )^{3/4} n}-\frac{2\ 2^{3/4} c^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-b+\sqrt{b^2-4 a c}}}\right )}{\sqrt{b^2-4 a c} \left (-b+\sqrt{b^2-4 a c}\right )^{3/4} n}\\ \end{align*}

Mathematica [A]  time = 0.878652, size = 340, normalized size = 0.96 \[ \frac{2\ 2^{3/4} c^{3/4} \left (-\frac{\sqrt [4]{-\sqrt{b^2-4 a c}-b} \tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{\sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}-\frac{\sqrt [4]{-\sqrt{b^2-4 a c}-b} \tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{-\sqrt{b^2-4 a c}-b}}\right )}{b \sqrt{b^2-4 a c}-4 a c+b^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt [4]{c} x^{n/4}}{\sqrt [4]{\sqrt{b^2-4 a c}-b}}\right )}{\sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}-b\right )^{3/4}}\right )}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + n/4)/(a + b*x^n + c*x^(2*n)),x]

[Out]

(2*2^(3/4)*c^(3/4)*(-(((-b - Sqrt[b^2 - 4*a*c])^(1/4)*ArcTan[(2^(1/4)*c^(1/4)*x^(n/4))/(-b - Sqrt[b^2 - 4*a*c]
)^(1/4)])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c])) - ArcTan[(2^(1/4)*c^(1/4)*x^(n/4))/(-b + Sqrt[b^2 - 4*a*c])^(1/
4)]/(Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4)) - ((-b - Sqrt[b^2 - 4*a*c])^(1/4)*ArcTanh[(2^(1/4)*c^(1
/4)*x^(n/4))/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(b^2 - 4*a*c + b*Sqrt[b^2 - 4*a*c]) - ArcTanh[(2^(1/4)*c^(1/4)*x
^(n/4))/(-b + Sqrt[b^2 - 4*a*c])^(1/4)]/(Sqrt[b^2 - 4*a*c]*(-b + Sqrt[b^2 - 4*a*c])^(3/4))))/n

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Maple [C]  time = 0.256, size = 280, normalized size = 0.8 \begin{align*} \sum _{{\it \_R}={\it RootOf} \left ( \left ( 256\,{a}^{7}{c}^{4}{n}^{8}-256\,{a}^{6}{b}^{2}{c}^{3}{n}^{8}+96\,{a}^{5}{b}^{4}{c}^{2}{n}^{8}-16\,{a}^{4}{b}^{6}c{n}^{8}+{a}^{3}{b}^{8}{n}^{8} \right ){{\it \_Z}}^{8}+ \left ( -48\,{a}^{3}b{c}^{3}{n}^{4}+40\,{a}^{2}{b}^{3}{c}^{2}{n}^{4}-11\,a{b}^{5}c{n}^{4}+{b}^{7}{n}^{4} \right ){{\it \_Z}}^{4}+{c}^{3} \right ) }{\it \_R}\,\ln \left ({x}^{{\frac{n}{4}}}+ \left ( 16\,{\frac{{n}^{5}b{a}^{5}{c}^{2}}{{c}^{2}a-{b}^{2}c}}-8\,{\frac{{n}^{5}{b}^{3}{a}^{4}c}{{c}^{2}a-{b}^{2}c}}+{\frac{{n}^{5}{b}^{5}{a}^{3}}{{c}^{2}a-{b}^{2}c}} \right ){{\it \_R}}^{5}+ \left ( 2\,{\frac{{a}^{2}{c}^{2}n}{{c}^{2}a-{b}^{2}c}}-4\,{\frac{a{b}^{2}cn}{{c}^{2}a-{b}^{2}c}}+{\frac{{b}^{4}n}{{c}^{2}a-{b}^{2}c}} \right ){\it \_R} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1+1/4*n)/(a+b*x^n+c*x^(2*n)),x)

[Out]

sum(_R*ln(x^(1/4*n)+(16/(a*c^2-b^2*c)*n^5*b*a^5*c^2-8/(a*c^2-b^2*c)*n^5*b^3*a^4*c+1/(a*c^2-b^2*c)*n^5*b^5*a^3)
*_R^5+(2/(a*c^2-b^2*c)*n*a^2*c^2-4/(a*c^2-b^2*c)*n*b^2*a*c+1/(a*c^2-b^2*c)*n*b^4)*_R),_R=RootOf((256*a^7*c^4*n
^8-256*a^6*b^2*c^3*n^8+96*a^5*b^4*c^2*n^8-16*a^4*b^6*c*n^8+a^3*b^8*n^8)*_Z^8+(-48*a^3*b*c^3*n^4+40*a^2*b^3*c^2
*n^4-11*a*b^5*c*n^4+b^7*n^4)*_Z^4+c^3))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/4*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="maxima")

[Out]

integrate(x^(1/4*n - 1)/(c*x^(2*n) + b*x^n + a), x)

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Fricas [B]  time = 4.15006, size = 9276, normalized size = 26.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/4*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="fricas")

[Out]

-2*sqrt(2)*sqrt(sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6
*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2
)*n^4)))*arctan(1/16*sqrt(2)*(2*sqrt(2)*((a^3*b^10*c - 15*a^4*b^8*c^2 + 86*a^5*b^6*c^3 - 232*a^6*b^4*c^4 + 288
*a^7*b^2*c^5 - 128*a^8*c^6)*n^7*x*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 -
64*a^9*c^3)*n^8)) - (b^9*c - 10*a*b^7*c^2 + 33*a^2*b^5*c^3 - 40*a^3*b^3*c^4 + 16*a^4*b*c^5)*n^3*x)*x^(1/4*n -
1)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c
+ 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)) + sqrt(2)*((
a^3*b^8 - 14*a^4*b^6*c + 72*a^5*b^4*c^2 - 160*a^6*b^2*c^3 + 128*a^7*c^4)*n^7*x*sqrt((b^4 - 2*a*b^2*c + a^2*c^2
)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - (b^7 - 9*a*b^5*c + 24*a^2*b^3*c^2 - 16*a^3*b
*c^3)*n^3*x)*sqrt((4*(b^4*c^2 - 2*a*b^2*c^3 + a^2*c^4)*x^2*x^(1/2*n - 2) - sqrt(2)*((a^3*b^9 - 13*a^4*b^7*c +
60*a^5*b^5*c^2 - 112*a^6*b^3*c^3 + 64*a^7*b*c^4)*n^6*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c
+ 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - (b^8 - 8*a*b^6*c + 21*a^2*b^4*c^2 - 22*a^3*b^2*c^3 + 8*a^4*c^4)*n^2)*s
qrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*
a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))/x^2)*sqrt(-((a^3
*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^
2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))*sqrt(sqrt(2)*sqrt(-((a^3*b
^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2
- 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))/(b^4*c^3 - 2*a*b^2*c^4 + a^2
*c^5)) + 2*sqrt(2)*sqrt(sqrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2
)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*
a^5*c^2)*n^4)))*arctan(1/8*(2*((a^3*b^10*c - 15*a^4*b^8*c^2 + 86*a^5*b^6*c^3 - 232*a^6*b^4*c^4 + 288*a^7*b^2*c
^5 - 128*a^8*c^6)*n^7*x*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^
3)*n^8)) + (b^9*c - 10*a*b^7*c^2 + 33*a^2*b^5*c^3 - 40*a^3*b^3*c^4 + 16*a^4*b*c^5)*n^3*x)*x^(1/4*n - 1)*sqrt(s
qrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*
c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))*sqrt(((a^
3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c
^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)) + ((a^3*b^8 - 14*a^4*b^6*c
+ 72*a^5*b^4*c^2 - 160*a^6*b^2*c^3 + 128*a^7*c^4)*n^7*x*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b
^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + (b^7 - 9*a*b^5*c + 24*a^2*b^3*c^2 - 16*a^3*b*c^3)*n^3*x)*sqrt(sqrt
(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c +
48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))*sqrt((4*(b^4
*c^2 - 2*a*b^2*c^3 + a^2*c^4)*x^2*x^(1/2*n - 2) + sqrt(2)*((a^3*b^9 - 13*a^4*b^7*c + 60*a^5*b^5*c^2 - 112*a^6*
b^3*c^3 + 64*a^7*b*c^4)*n^6*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^
9*c^3)*n^8)) + (b^8 - 8*a*b^6*c + 21*a^2*b^4*c^2 - 22*a^3*b^2*c^3 + 8*a^4*c^4)*n^2)*sqrt(((a^3*b^4 - 8*a^4*b^2
*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*
n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))/x^2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5
*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3
+ 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))/(b^4*c^3 - 2*a*b^2*c^4 + a^2*c^5)) + 1/2*sqrt(2)*sqrt
(sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b
^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))*log(-(
4*(b^2*c - a*c^2)*x*x^(1/4*n - 1) + sqrt(2)*((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*n^5*sqrt((b^4 - 2*a*b^2*c
+ a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - (b^4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sq
rt(sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7
*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4))))/x)
- 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((
a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*
c^2)*n^4)))*log(-(4*(b^2*c - a*c^2)*x*x^(1/4*n - 1) - sqrt(2)*((a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*n^5*sqrt
((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - (b^4 - 5*a*b^2*c
+ 4*a^2*c^2)*n)*sqrt(sqrt(2)*sqrt(-((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/
((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) + b^3 - 3*a*b*c)/((a^3*b^4 - 8*a^4*b^2*c + 16*a^
5*c^2)*n^4))))/x) - 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b^
2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a^
4*b^2*c + 16*a^5*c^2)*n^4)))*log(-(4*(b^2*c - a*c^2)*x*x^(1/4*n - 1) + sqrt(2)*((a^3*b^5 - 8*a^4*b^3*c + 16*a^
5*b*c^2)*n^5*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) +
(b^4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sqrt(sqrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sqrt((b^4 - 2*a*b
^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/((a^3*b^4 - 8*a
^4*b^2*c + 16*a^5*c^2)*n^4))))/x) + 1/2*sqrt(2)*sqrt(sqrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*sq
rt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)/
((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4)))*log(-(4*(b^2*c - a*c^2)*x*x^(1/4*n - 1) - sqrt(2)*((a^3*b^5 - 8*a
^4*b^3*c + 16*a^5*b*c^2)*n^5*sqrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a
^9*c^3)*n^8)) + (b^4 - 5*a*b^2*c + 4*a^2*c^2)*n)*sqrt(sqrt(2)*sqrt(((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4*s
qrt((b^4 - 2*a*b^2*c + a^2*c^2)/((a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*n^8)) - b^3 + 3*a*b*c)
/((a^3*b^4 - 8*a^4*b^2*c + 16*a^5*c^2)*n^4))))/x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{n}{4} - 1}}{a + b x^{n} + c x^{2 n}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+1/4*n)/(a+b*x**n+c*x**(2*n)),x)

[Out]

Integral(x**(n/4 - 1)/(a + b*x**n + c*x**(2*n)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{\frac{1}{4} \, n - 1}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1+1/4*n)/(a+b*x^n+c*x^(2*n)),x, algorithm="giac")

[Out]

integrate(x^(1/4*n - 1)/(c*x^(2*n) + b*x^n + a), x)